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Abstract

The persistence properties of solutions to the dissipative 2-component Degasperis-Procesi
system are investigated. We find that if the initial data with their derivatives of
the system exponentially decay at infinity, then the corresponding solution also exponentially
decays at infinity.

Keywords:

1 Introduction

We consider the following dissipative 2-component Degasperis-Procesi system:

(1.1)

where λ, are nonnegative constants, , ().

In system (1.1), if , we get the classical Degasperis-Procesi equation [1]

(1.2)

where represents the fluid velocity at time t in x direction and . The nonlinear convection term causes the steepening of wave form. The nonlinear dispersion effect term makes the wave form spread. The Degasperis-Procesi equation has been studied in many
works [2-8]. Escher et al.[2] demonstrated that there exists a unique solution to (1.2) with initial value (). Liu and Yin [3] obtained the global existence of solutions to (1.2). They derived several wave breaking
mechanisms in Sobolev space with . Yin [4] established the local well-posedness for the Degasperis-Procesi equation with initial
value () on the line. In [5], the author obtained the global existence of solutions to the Degasperis-Procesi
equation on the circle. The precise blow-up scenario was also derived. The global
existence of strong solutions and global weak solutions to the Degasperis-Procesi
equation were shown in [6,7]. Guo et al.[9] studied the dissipative Degasperis-Procesi equation,

(1.3)

where () is the dissipative term. They obtained the global weak solutions to (1.3). Guo [10] established the local well-posedness for (1.3), and also obtained the global existence,
persistence properties and propagation speed of solutions. Wu and Yin [8] obtained the local well-posedness for (1.3), and also studied the blow-up scenarios
of solutions in periodic case.

On the other hand, many researchers have studied the integrable multi-component generalizations
of the Degasperis-Procesi equation [11-16]. Yan and Yin [11] investigated the 2-component Degasperis-Procesi system

(1.4)

where . They established the local well-posedness for system (1.4) in Besov space with , and also derived the precise blow-up scenarios of strong solutions in Sobolev space
with . Zhou et al.[12] investigated the traveling wave solutions to the 2-component Degasperis-Procesi system.
Manwai [16] studied the self-similar solutions to the 2-component Degasperis-Procesi system.
Fu and Qu [13] obtained the persistence properties of solutions to the 2-component Degasperis-Procesi
system in Sobolev space with . For system (1.4), Jin and Guo [14] studied the blow-up mechanisms and persistence properties of strong solutions.

Recently, a large amount of literature has been devoted to the study of the 2-component
Camassa-Holm system [17-28]. Hu [18] studied the dissipative periodic 2-component Camassa-Holm system

(1.5)

where . The author not only established the local well-posedness for system (1.5) in Besov
space with , but she also presented global existence results and the exact blow-up scenarios
of strong solutions in Sobolev space with . For in system (1.5), Jin and Guo [19] considered the persistence properties of solutions to the modified 2-component Camassa-Holm
system. Zhu [20] considered the persistence property of solutions to the coupled Camassa-Holm system,
and also established the global existence and blow-up mechanisms of solutions. Guo
[21,22] studied the persistence properties and unique continuation of solutions to the 2-component
Camassa-Holm system in the case . It was shown in [29] that the dissipative Camassa-Holm, Degasperis-Procesi, Hunter-Saxton and Novikov
equations could be reduced to the non-dissipative versions by means of an exponentially
time-dependent scaling. One may refer to [30-34] and the references therein for more details in this direction.

Motivated by the work in [13,20,35], we study the dissipative 2-component Degasperis-Procesi system (1.1). We note that
the persistence properties of solutions to system (1.1) have not been discussed yet.
The aim of this paper is to investigate the persistence properties of solutions in
Sobolev space . The main idea of this work comes from [35].

Now we rewrite system (1.1) as

(1.6)

where the operator .

The main results are presented as follows.

Theorem 1.1Assumeandwith. Then the Cauchy problem (1.1) has a unique solution.

Theorem 1.2Letandwith. is the corresponding solution to system (1.1). If there existssuch that

then

uniformly on the interval.

Theorem 1.3Letandwith. is the corresponding solution to system (1.1). Assume the constant.

(1) For,

and there existssuch thatas, then

(2) For,

and there existssuch thatas, then

Theorem 1.4Letin system (1.1). Assumeandwith. is the corresponding solution to system (1.1). If there existssuch that

then

uniformly on the interval.

Theorem 1.5Assumeandwith. is the corresponding solution to system (1.1). If there existssuch that

then

Theorem 1.6Assumeandwith. is the corresponding solution to system (1.1). If there existssuch that

and there existssuch that

then

The remainder of this paper is organized as follows. In Section 2, the proofs of Theorems 1.1
and 1.2 are presented. Section 3 is devoted to the proofs of Theorems 1.3 and 1.4.
The proofs of Theorems 1.5 and 1.6 are given in Section 4.

Notation We denote the norm of Lebesgue space , by , the norm in Sobolev space , by and the norm in Besov space , by . For , we denote

2 Proofs of Theorems 1.1 and 1.2

We write the definition of Besov space. One may check [36-39] for more details.

2.1 Proof of Theorem 1.1

Using the Littlewood-Paley theory and estimates for solutions to the transport equation,
one may follow similar arguments as in [11] to establish the local well-posedness for system (1.1) with some modification. Here
we omit the detailed proof. For system (1.1) with initial data (), we see that the corresponding solution . Thus we complete the proof of Theorem 1.1.

2.2 Proof of Theorem 1.2

We denote

Multiplying the second equation in (1.6) by with and integrating the resultant equation with respect to x yield

(2.1)

We have

Thus

(2.2)

If , using the Sobolev embedding theorem, we have . Applying the Gronwall inequality to (2.2) yields

Noting gives rise to

and, taking the limit as , we obtain

Multiplying the first equation in system (1.6) by with and integrating the resultant equation with respect to x yield

(2.3)

Using the Holder inequality, we have

(2.4)

which in combination with (2.3) yields

Using the Gronwall inequality, one derives

(2.5)

Taking the limit as in (2.5), one gets

(2.6)

Differentiating the first equation in (1.6) in the variable x yields

(2.7)

Multiplying (2.7) by with , integrating the resultant equation with respect to x and using

(2.8)

and

we have

We obtain

(2.9)

We introduce the weight function which is independent on t

where . It follows a.e. . Multiplying the first equation in system (1.6) and (2.7) by , we obtain

(2.10)

(2.11)

Multiplying (2.10) by and (2.11) by , respectively, and integrating the resultant equation with respect to x, we also note

As in the weightless case, we estimate and step by step as the previous estimates for u and . Thus

(2.12)

Multiplying the second equation in system (1.6) by , one deduces

(2.13)

Multiplying (2.13) by , integrating the resultant equation with respect to x and using

we have

Applying the Gronwall inequality and the Sobolev embedding theorem yields

Taking the limit as , one obtains

(2.14)

There exists which depends on , such that for all

Thus

(2.15)

Using for all f, we have

(2.16)

Plugging (2.15), (2.16) into (2.12) and using (2.14), there exists such that

(2.17)

Using the Gronwall inequality, one deduces that for all and

Finally, taking the limit as , one obtains

(2.18)

Thus

uniformly on the interval . This completes the proof of Theorem 1.2.

3 Proofs of Theorems 1.3 and 1.4

3.1 Proof of Theorem 1.3

(1) For , integrating the first equation in (1.6) over the interval , one has

(3.1)

From the assumption in Theorem 1.3, one deduces

(3.2)

It follows from Theorem 1.2 that

For , we have

For the right side in (3.1), we have

(3.3)

where . From Theorem 1.2, one has

Then

(3.4)

Noting , if there is at least one of the equalities and is valid, we have . Then there exists such that

Thus

which combined with the above estimates yields a contradiction. We obtain . Consequently, , .

(2) For , similar to the case , one deduces . Inserting into the second equation in (1.1), one derives

(3.5)

From (3.5), we have . This completes the proof of case (2) in Theorem 1.3.

3.2 Proof of Theorem 1.4

For , integrating the first equation in system (1.6) on the interval , one obtains

(3.6)

From the assumption in Theorem 1.4 as and Theorem 1.2, one deduces

(3.7)

For , then

For the right side in (3.6), firstly, we have

(3.8)

where . Using Theorem 1.2, we obtain

Thus

(3.9)

Noting

and

we have

Similarly, we have

Then as . From Theorem 1.2, if as , we have as . This completes the proof of Theorem 1.4.

4 Proofs of Theorems 1.5 and 1.6

4.1 Proof of Theorem 1.5

From the proof of Theorem 1.2, here we need to differentiate the second equation in
(1.6) with variable x, and one has

(4.1)

Multiplying (4.1) by and integrating the resultant equation with respect to x, we also note

Thus, we obtain

(4.2)

Taking the limit as and applying the Gronwall inequality yield

(4.3)

In order to obtain the estimates for , we multiply (4.1) with the weight function , then

(4.4)

Multiplying (4.4) with and integrating the resultant equation with respect to x, we note

Hence, we have

(4.5)

Taking the limit as and using the Gronwall inequality, one obtains

(4.6)

From (4.6) and (2.17), one deduces that there exists such that

(4.7)

where

Applying the Gronwall inequality to (4.7), for all and , one has

(4.8)

Now taking the limit as in (4.8), one obtains

Using the assumption in Theorem 1.5, we complete the proof.

4.2 Proof of Theorem 1.6

The proof of Theorem 1.6 is similar to the proof of Theorem 1.3, here we omit it.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed to each part of this study equally and approved the final
version of the manuscript.

Acknowledgements

The authors would like to express sincere gratitude to the anonymous referees for
a number of valuable comments and suggestions. This work was partially supported by
National Natural Science Foundation of P.R. China (71003082) and Fundamental Research
Funds for the Central Universities (SWJTU12CX061 and SWJTU09ZT36).